She's powerful, but her early game is so abysmal you almost can't justify her godly late game.

Top is dominated by super strong early game characters that have moderately good late games. Mid is almost always a ranged mage. Jungle is incredibly lacking for farm. Bottom lane as a carry? Unlikely to work, and as a support won't get the money anyway.

Where do you send her that doesn't turn the game into a 40 minute 4v5?

I mean, I've had some success with her top with lifesteal runes, but that's mostly because of people forgetting that killing Poppy from full under her tower is not going to happen.